This paper appears as an appendix to a chapter in a book by E. F. beckenbach entitled, 'Mathematics for Engineers.' This appendix sketches briefly some of the most important aspects of a part of the general theory of differential equations, the stability theory of equilibrium states.

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DIFFERENTIAL EQUATIONS, DIFFERENCE EQUATIONS, NONLINEAR...

The theory of time scales was introduced by Stefan Hilger in his 1988 PhD dissertation. Scholars in the fields of differential equations and difference equations have long been aware of the startling similarities and intriguing differences between the two fields. The study of dynamic equations on time scales unifies and extends the fields of differential and difference equations, highlighting the similarities and providing insight into some of the differences.

Topics: DTIC Archive, Messer, Kirsten R, AIR FORCE INST OF TECH WRIGHT-PATTERSONAFB OH, *DIFFERENTIAL...

It is shown that Laplace transform can be effective in the numerical solution of nonlinear functional equations. For illustrative purposes a nonlinear differential equation, a nonlinear differential-difference equation and a nonlinear diffusion equation are considered.

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *NONLINEAR DIFFERENTIAL EQUATIONS, *NUMERICAL METHODS AND...

This research centered around three topics in computational fluid dynamics. These were: schemes for the time dependent Navier Stokes equations, improved methods for the steady Navier-Stokes equations, and domain decomposition methods. (In the original proposal, the domain decomposition methods were referred to as overlapping grid methods.) All of these topics are coupled to the development faster algorithms for solving the systems of indefinite linear equations that arise from discretizations...

Topics: DTIC Archive, Strikwerda, John C, WISCONSIN UNIV-MADISON, *COMPUTATIONS, *NAVIER STOKES EQUATIONS,...

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DYNAMIC PROGRAMMING, *EQUATIONS, *INTEGRAL EQUATIONS,...

This research results are contained in the eleven papers published during the course of the research. Abstracts of these papers are contained in this document. Contents: Numerical Solutions to Stochastic Difference Equations; Approximate Solution of Random Differential Equations; Numerical Solution to a System of Random Volterra Integral Equations; Optimal Processes Governed by Integral Equations with Unilateral Constraint; Numerical Solution of Random Love's Integral Equation; Mixed Min-Max...

Topics: DTIC Archive, Bota, Kofi B, ATLANTA UNIV GA, *COMPUTATIONS, COEFFICIENTS, DIFFERENCE EQUATIONS,...

An analysis is presented of the torsional oscillations of an artillery barrel when subjected to the moment of an accelerating projectile. A modal series is obtained, the coefficients of which are the time dependent generalized coordinates. OR PURPOSES OF OBTAINING NUMERICAL RESULTS( THIS SERIES WAS TRUNCATED at two terms; the subsequent system of ordinary differential equations was programmed for the analog computer. Graphic solutions are given for several values of the barrel size to...

Topics: DTIC Archive, Smith, T, ILLINOIS UNIV AT URBANA-CHAMPAIGN, *GUN BARRELS, DIFFERENTIAL EQUATIONS,...

In this paper, the problem was considered of determining the asymptotic behavior of solutions of linear differentialdifference equations whose coefficients possess asymptotic series. Although the problem is considerably more complicated than the corresponding problem for ordinary differential equations, by means of a sequence of transformations the problem was reduced to a form where the standard techniques of ordinary differential equation theory could be employed. The differential-difference...

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DIFFERENTIAL EQUATIONS, *SERIES(MATHEMATICS), DIFFERENCE...

Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional V is constructed for a linear Volterra intergrodifferential equation. This functional, and its derivative, is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for...

Topics: DTIC Archive, Abrahamson,D L, BROWN UNIV PROVIDENCE RI LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS,...

Topics: DTIC Archive, Freeman, B E, GENERAL DYNAMICS SAN DIEGO CA GENERAL ATOMIC DIV, *REACTION KINETICS,...

This paper develops an algorithmic method for transforming quasilinear partial differential equations of a certain form into semilinear equations. This crucially involves the use of hodograph transformations (i.e., transformations which involve the interchange of dependent and independent variables). Furthermore, we find the most general quasilinear equation of the above form which can be mapped via a hodograph transformation to a semilinear form. This algorithm provides a method for...

Topics: DTIC Archive, Clarkson, P A, CLARKSON UNIV POTSDAM NY DEPT OF MATHEMATICS AND COMPUTER SCIENCE,...

A multigrid method is defined as having textbook multigrid efficiency (TME) if solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. Away to achieve TME for the Euler and Navier-Stokes equations is to apply the distributed relaxation method thereby separating the elliptic and hyperbolic partitions of the equations. Design of a distributed relaxation scheme can be...

Topics: DTIC Archive, Diskin, Boris, INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON...

References describe a new method of asymptotic integration of linear hyperbolic partial differential equations and show the application of this method for finding asymptotic solutions of acoustic and Maxwellian equations. In references 5-8, the indicated method is developed as it applies to the solution of dynamic problems of the theory of elasticity.

Topics: DTIC Archive, Skuridin, G. A., JOINT PUBLICATIONS RESEARCH SERVICE ARLINGTON VA, *PLASMAS(PHYSICS),...

Approximate separable representations of Green's functions for differential operators is a basic and an important aspect in the analysis of differential equations and in the development of efficient numerical algorithms for solving them. Being able to approximate a Green's function as a sum with few separable terms is equivalent to the existence of low rank approximation of corresponding discretized system. This property can be explored for matrix compression and efficient numerical algorithms....

Topics: DTIC Archive, CALIFORNIA UNIV LOS ANGELES DEPT OF MATHEMATICS, *GREENS FUNCTIONS, ALGORITHMS,...

Unlike other standard equations in introductory classical mechanics, the Bernoulli equation is not Galilean invariant. The explanation is that, in a reference frame moving with respect to constrictions or obstacles, those surfaces do work on the fluid, constituting an extra term that needs to be included in the work?energy calculation. A quantitative example is presented here for a horizontal tapered pipe. A frame-independent expression for the pressure drop in the pipe is obtained. The...

Topics: DTIC Archive, NAVAL ACADEMY ANNAPOLIS MD, *EQUATIONS, FLUIDS, INVARIANCE, MECHANICS, NONLINEAR...

Structural synthesis is the analysis of the dynamic response of a system when either subsystems are combined (substructure coupling) or modifications are made to substructures (structural modification). The integral equation formulation for structural synthesis is a method that requires only the baseline transient response, the baseline modal parameters, and the impedance of the structural modification. The integral formulation results in a Volterra integral equation of the second-kind. An...

Topics: DTIC Archive, NAVAL POSTGRADUATE SCHOOL MONTEREY CA, *INTEGRAL EQUATIONS, DYNAMIC RESPONSE,...

We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two dimensions. The algorithm is fast in the sense that its asymptotic complexity is O(NlogkN), where N is the number of nodes in the discretization, and K depends on the kernel and the geometry of the contour (k= 1 or 2). Unlike previous fast techniques based on iterative solvers, the present algorithm...

Topics: DTIC Archive, YALE UNIV NEW HAVEN CT DEPT OF MATHEMATICS, *ALGORITHMS, *INTEGRAL EQUATIONS, LINEAR...

The study of dynamic programming processes of continuous type gives rise to functional equations of the form dx/dt = Max/q f (x,t;q). In this paper we present a summary of some basic results concerning the existence and uniqueness of solutions of this equation. Detailed results will appear subsequently.

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DYNAMIC PROGRAMMING, *FUNCTIONS(MATHEMATICS), *NONLINEAR...

Topics: DTIC Archive, BELLMAN, RICHARD, RAND CORP SANTA MONICA CA, *EQUATIONS, *PERTURBATION THEORY,...

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DYNAMIC PROGRAMMING, BOUNDARY VALUE PROBLEMS, EQUATIONS,...

In this paper, we investigate linear relations among the Euler function of nearby integers. In particular, we study those positive integers n such that theta(n) = theta(n -1) + (n - 2), where theta is the Euler function. We prove that they form a set of asymptotic density zero. We also show that the sum of the reciprocals of the prime values of n with the above property is a convergent series.

Topics: DTIC Archive, NAVAL POSTGRADUATE SCHOOL MONTEREY CA DEPT OF APPLIED MATHEMATICS, *LINEAR ALGEBRAIC...

It is shown that the functional equation technique of the theory of dynamic programming may be used to derive functional differential equations for the characteristic values of a certain integral equation similar to those obtained for the eigenvalues of differential equations.

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *DYNAMIC PROGRAMMING, *INTEGRAL EQUATIONS, DIFFERENTIAL...

Two papers are given, their purpose being to study in detail some representative problems which occur in the study of interindustry processes. The general techniques will be those of the theory of dynamic programming. The first paper considers a simple mathematical model of the interaction of two independent industries engaged in the production of one item. For convenience, the auto and steel industries are considered. The second paper treats the problem of maximizing x sub 2 (T) given the...

Topics: DTIC Archive, BELLMAN, RICHARD, RAND CORP SANTA MONICA CA, *COMPUTER PROGRAMMING, *GAME THEORY,...

Periodic solutions of the equations of motion of a satellite of an oblate planet are investigated, using methods developed by Poincare (1892). The treatment covers both critical and non-critical angles.

Topics: DTIC Archive, BARRAR, RICHARD B, SYSTEM DEVELOPMENT CORP SANTA MONICA CA, *SYNCHRONOUS SATELLITES,...

It is difficult to predict stability properties of a finite difference scheme. It has to be investigated through the roots of the Z-transformed and Fourier transformed difference scheme (modal equation). To simultaneously investigate several schemes for the viscoelastic wave equation, it is possible to derive the modal equation with parameterized coefficients. Several conditionally stable schemes were found, where the most efficient is a staggered scheme with a stability condition closely...

Topics: DTIC Archive, Blanch, Joakim O, RICE UNIV HOUSTON TX DEPT OF COMPUTATIONAL AND APPLIED MATHEMATICS,...

A major thrust of the original proposal was to find new probabilistic methods for dealing with semilinear partial differential equations. Mathematicians are currently devoting more of their attention to studying nonlinear partial differential equations since they recognize that descriptions of physical phenomena must incorporate nonlinear behavior. The author succeeded in finding a method for solving systems of semilinear elliptic equations by a new procedure which does not need the old...

Topics: DTIC Archive, Glover, Joseph, FLORIDA UNIV GAINESVILLE, *PROBABILITY, *PARTIAL DIFFERENTIAL...

There is strong physical evidence that a full treatment of differential-algebraic equations should be incorporate solutions with jump discontinuities. It is shown here that for semilinear problems the setting of distributions allows for the development of a theory where indeed such discontinuities may occur. This approach also settles the problem of inconsistent initial conditions in a very simple way. On the other hand, new issues arise as not only uniqueness. but even countability of the...

Topics: DTIC Archive, Rabier, Patrick J., PITTSBURGH UNIV PA DEPT OF MATHEMATICS AND STATISTICS,...

The equations of motion of a ship constrained to oscillate in heave, pitch, and roll, while moving ahead with a uniform velocity in oblique waves are studied. It is shown that because of the nonlinear static coupling which is known to exist between these degrees of freedom, the rolling response can become large when the period of wave encounter is in the neighborhood of one half the natural rolling period. In this case the response has a period which is equal to twice the period of the...

Topics: DTIC Archive, Kinney, W D, CALIFORNIA UNIV BERKELEY INST OF ENGINEERING RESEARCH, *SHIPS, ROLL,...

This final report deals primarily with the study of infinite dimensional dynamical systems and, more specifically, with hyperbolic and parabolic partial differential equations. Part of the final report is devoted to showing that a nice dynamical system is defined for specific types of equations that occur often in models for physical systems. The remainder of the report belongs to the general category of the investigation of the qualitative properties of infinite dimensional flows; in...

Topics: DTIC Archive, Hale, Jack, BROWN UNIV PROVIDENCE RI DIV OF APPLIED MATHEMATICS, *DYNAMICS, *PARTIAL...

In recent years considerable interest has focused on certain physically important nonlinear evolution equations which can be linearized. Many of these equations fall into the category of linearization via soliton theory and the Inverse Scattering Transform (IST). Well-known equations are the Korteweg-deVries Equation (KdV), the sine-Gordon equation, and the Kadomtsev-Petviashvili (KP) equation. Each of these equations has certain singular integro-differential analogs, the best known being the...

Topics: DTIC Archive, Ablowitz, M J, CLARKSON UNIV POTSDAM NY INST FOR NONLINEAR STUDIES, *INTEGRAL...

An overview of a normal mode method of solving the Helmholtz wave equation to describe the underwater sound field for a fixed point source in a plane multilayered medium is presented. The mode functions are well-defined at all depths of the medium as they are continuous across turning points of the separated depth-dependent differential equation. Comparisons of model results to a limited number of benchmark propagation solutions are presented.

Topics: DTIC Archive, NAVAL UNDERSEA WARFARE CENTER DIV NEWPORT RI, *WAVE PROPAGATION, *HELMHOLTZ...

A linearized theory for the unsteady motion of a partially cavitated flat hydrofoil in two dimensions is carried out. A second linearization procedure is used, based on ideas of Timman and Guerst, to obtain the unsteady pressure distribution around the hydrofoil and the resulting force and moment, as functions of the cavitation number and Strouhal number for given pitch and/or heave.

Topics: DTIC Archive, Steinberg, Herbert, CONTROL DATA CORP MELVILLE NY TRG DIV, *HYDROFOILS, *CAVITATION,...

New fast time domain integral equation (TDIE) solvers for analyzing large-scale electromagnetic scattering and radiation phenomena were developed. These solvers were accelerated using a host of new Plane Wave Time Domain (PWTD) algorithms. PWTD augmented TDIE solvers accomplish for the time domain wave equation (and related Maxwell's equations) what fast multipole methods previously achieved for the Poisson and Helmholtz equations, viz. a significant reduction in computational cost of the...

Topics: DTIC Archive, Michielsssen, Eric, ILLINOIS UNIV AT URBANA BOARD OF TRUSTEES, *ELECTROMAGNETIC...

The essential goal of this investigation was to combine symmetry-reduction techniques with methods of nonlinear analysis to solve nonlinear differential equations associated with problems from elastomechanics. During the effective time of this grant papers have appeared and papers have been accepted for publication.

Topics: DTIC Archive, Healey, Timothy J, CORNELL UNIV ITHACA NY CENTER FOR APPLIED MATHEMATICS, *ELASTIC...

By utilizing the Euler-Lagrange equations, a set of coupled partial differential equations was derived for two crossing strings with a spring at the crossover point. This report considers three types of springs: a nonlinear softening spring, a nonlinear hardening spring, and a linear spring. The Adomian decomposition method was used to obtain an analytical approximate solution from the derived coupled wave equations. The dynamic responses of the analytical solutions for both the nonlinear...

Topics: DTIC Archive, NAVAL UNDERSEA WARFARE CENTER DIV NEWPORT RI RANGES ENGINEERING AND ANALYSIS DEPT,...

We introduce a method for constructing a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomial-based NLLS-optimized (PBNO) preconditioner significantly improves the performance of 2-D continuous Galerkin (CG) and discontinuous Galerkin (DG) fluid dynamical research models when run in an implicit-explicit time integration mode. When employed in a serially computed Schur-complement form of the 2-D CG model with positive definite spectrum, the...

Topics: DTIC Archive, NAVAL POSTGRADUATE SCHOOL MONTEREY CA DEPT OF APPLIED MATHEMATICS, *ATMOSPHERE...

The research project was concerned with the systematic development of computational methods for random equations. An earlier ARO research project was concerned primarily with the development of computational methods for the solution of random integral equations. This project was concerned with the computational solution of random integral equations as well as other classes of random equations, with special reference to computer implementation of general methods for obtaining approximate...

Topics: DTIC Archive, Bharucha-Reid,A T, GEORGIA INST OF TECH ATLANTA SCHOOL OF MATHEMATICS, *Equations,...

This final report summarizes the principal investigators' achievements on the research project during the period September 1, 1987 through November 30, 1989. These include new results for wave equations and plate equations, linear and nonlinear, on the following problems: exact controllability, strong and uniform stabilization, structural damping, quadratic optimal control problem, Riccati equations, robustness with respect to nonlinear uncertainties, and numerical aspects thereof. (sdw)

Topics: DTIC Archive, Triggiani, R, VIRGINIA UNIV CHARLOTTESVILLE DEPT OF APPLIED MATHEMATICS, *STRUCTURAL...

This document shows that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the L sub 1 norm of the initial data. This implies that the dissipation due to the entropy dominates the nonlinearities in the problem at a rate depending only on the L sub 1 norm of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate...

Topics: DTIC Archive, Temple,B, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, NONLINEAR DIFFERENTIAL...

Topics: DTIC Archive, Londen,Stig-Olof, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, *VOLTERRA...

Topics: DTIC Archive, Vanden-Broeck,Jean-Marc, STANFORD UNIV CA, *NONLINEAR DIFFERENTIAL EQUATIONS,...

In this paper a detailed study of the stability properties of the quadratic differential equation is undertaken. After observing that such systems can never be asymptotically stable the equilibrium states of the quadratic differential equation are classified in terms of the matrices G and H. Necessary and sufficient conditions for the stability of the origin are derived and constitute the principal contribution of this paper. Finally, these conditions are re-derived and elaborated using polar...

Topics: DTIC Archive, Koditschek,Daniel E, YALE UNIV NEW HAVEN CONN SYSTEMS AND INFORMATION SCIENCES,...

The objective of this language, which we call PDELAN, is to facilitate the coding of finite difference schemes for partial differential equations. The aspect of these codes which we have emphasized is the difference equations. An operator notation is provided so that the equations can be written as the numerical analyst frequently invites them prior to translation into a program.

Topics: DTIC Archive, Gary, John, COLORADO UNIV AT BOULDER DEPT OF COMPUTER SCIENCE, *NUMERICAL ANALYSIS,...

One of the basic problems of mathematical physics is that of replacing a nonlinear functional equation by a more tractable (analytically and computationally) linear equation. More generally, one wants to replace a system of nonlinear equations, often derived from a single equation by means of an expansion in a parameter, an ortho gonal expansion with a system of linear functional equations. To treat this closure problem, the authors presented a method based upon the concept of relative...

Topics: DTIC Archive, RAND CORP SANTA MONICA CA, *INVARIANCE, BOUNDARY VALUE PROBLEMS, DIFFERENTIAL...

A set of coupled integral equations is derived from the incompressible Navier-Stokes equations and the continuity equation. These equations are based on a vorticity-velocity-enthalpy formulation and are exact. The equations consist of a generalization of the Biot-Savart law for determining the velocity, an integral expression of the momentum equation for determining the vorticity, and a boundary integral equation for determining the stagnation enthalpy. The equations are linear in each...

Topics: DTIC Archive, Uhlman, J S , Jr, NAVAL UNDERSEA WARFARE CENTER DIV NEWPORT RI, *EQUATIONS OF MOTION,...

We consider the motion of a viscous fluid filling the whole space R3, governed by the classical Navier-Stokes equations (1). Existence of global (in time) regular solutions for that system of non-linear partial differential equations, is still an open problem. From either the mathematical and the physical point of view, an interesting property is the stability (or not) of the (eventual) global regular solutions. Here, we assume that v1(t,x) is a solution, with initial data a1(x). For small...

Topics: DTIC Archive, Beirao da Veiga,H, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, *NAVIER STOKES...

In a recent paper Chen and Saffman showed that periodic gravity waves in water of infinite depth are not unique. They presented explicit computations for new families of waves which they termed irregular waves. In the present paper the authors give conclusive numerical evidence that periodic gravity waves in water of arbitrary uniform depth are not unique. Explicit computations of irregular waves in water of finite depth are presented. In addition, they show that Chen and Saffman's bifurcation...

Topics: DTIC Archive, Vanden-Broeck,Jean-Marc, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, *WATER,...

This paper considers the problem of stabilizing a control system governed by a combination of partial and ordinary differential equations. The partial differential equations govern the evolution of the system in the interior of some spatial domain, the ordinary differential equations describe the evolution of the boundary data; the control enters through the boundary ordinary differential equations in a bilinear fashion. We provide sufficient conditions for feedback stabilization of such...

Topics: DTIC Archive, Slemrod,M, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, *Partial differential...

We consider the motion of a viscous fluid filling the whole space R3, governed by the classical Navier-Stokes equations (1). Existence of global (in time) regular solutions for that system of non-linear partial differential equations is still an open problem. Up to now, the only available global existence theorem (other than for sufficiently small initial data) is that of weak (turbulent) solutions. From both the mathematical and the physical point of view, an interesting property is the...

Topics: DTIC Archive, Secchi,Paolo, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER, *NAVIER STOKES...

Some general problems are described concerning the asymptotic behavior of control processes as the time-interval becomes infinite. Some partial results in the general case, and a detailed analysis of a one- dimensional control process are presented.

Topics: DTIC Archive, Bellman, Richard, RAND CORP SANTA MONICA CA, *CALCULUS OF VARIATIONS, BOUNDARY VALUE...